Polynomial phases for multi-carrier modulation schemes with time domain windowing

ABSTRACT

In one aspect, a method includes performing a mapping on bits to form a complex data symbol, applying a frequency rotation mask to the complex data symbol based on a polynomial phase, performing an inverse discrete Fourier transform (IDFT) after applying the frequency rotation mask, applying a time domain window after performing the IDFT, converting digital data to analog data after applying the time window and transmitting the analog data as an analog signal.

BACKGROUND

Multi-carrier modulation (MCM) is a method of transmitting a bit streamby dividing the bit stream into multiple components and transmittingeach component over a separate carrier signal or frequency (also calleda subcarrier) from the other components. Orthogonal frequency divisionmultiplexing (OFDM) is an MCM technique used to provide robust signalingcapabilities in complex environments where the channel transfer functioncan exhibit significant frequency dependency. In addition, OFDM providesa simple method for arbitrary fragmentation of an available channel or aconsecutive chunk of bandwidth.

SUMMARY

In one aspect, a method includes performing a mapping on bits to form acomplex data symbol, applying a frequency rotation mask to the complexdata symbol based on a polynomial phase, performing an inverse discreteFourier transform (IDFT) after applying the frequency rotation mask,applying a time domain window after performing the IDFT, convertingdigital data to analog data after applying the time window andtransmitting the analog data as an analog signal.

In another aspect, an apparatus includes electronic hardware circuitryconfigured to perform a mapping on bits to form a complex data symbol,apply a frequency rotation mask to the complex data symbol based on apolynomial phase, perform an inverse discrete Fourier transform (IDFT)after applying the frequency rotation mask, apply a time domain windowafter performing the IDFT, convert digital data to analog data afterapplying the time window and transmit the analog data as an analogsignal.

In a further aspect, an article includes a non-transitorycomputer-readable medium that stores computer-executable instructions.The instructions causing a machine to perform a mapping on bits to forma complex data symbol, apply a frequency rotation mask to the complexdata symbol based on a polynomial phase, perform an inverse discreteFourier transform (IDFT) after applying the frequency rotation mask,apply a time domain window after performing the IDFT, convert digitaldata to analog data after applying the time window and transmit theanalog data as an analog signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a functional block diagram of an example of a system totransmit data.

FIG. 1B is a functional block diagram of an example of a system toreceive data transmitted by the system of FIG. 1.

FIG. 2 is a flowchart of an example of a process to transmit data.

FIG. 3 is a flowchart of an example of a process to receive data.

FIGS. 4A to 4C are graphs of fast Fourier transform (FFT) frames andreal and imaginary time domain data without using a frequency rotationmask.

FIG. 4D to 4F are graphs of FFT frames and real and imaginary timedomain data using a frequency rotation mask.

FIGS. 5A to 5C are graphs comparing Newman, Narahashi-Nojima andpolynomial frequency rotation masks.

FIG. 6A is a graph using a Newman mask transmitting all 1's before andafter a time-domain window.

FIG. 6B is a graph using a polynomial mask transmitting all 1's beforeand after a time-domain window.

FIG. 7 is a computer on which the processes of FIGS. 2 and 3 may beimplemented.

DETAILED DESCRIPTION

Described herein are techniques that use a frequency rotation mask,which ensures that a maximum sub-carrier to sub-carrier power deviationis within +/−5 dB for any arbitrary data pattern. In particular thetechniques include a selection of the frequency rotation mask and theinterplay of the frequency rotation mask with a windowing function usedto mitigate out of band radiation.

The use of windowing functions to improve the spectral containment ofOFDM waveforms is very attractive, but it often presents significantproblems when applied to data streams that result in high Peak toAverage Power Ratio (PAPR) OFDM frames. The result of windowing theseframes is that a large number of OFDM sub-carriers in the transmittedsignal will be attenuated significantly. The techniques described hereincontrol the PAPR and reduce the negative effects of sub-carrierattenuation while allowing the use windowing functions.

Referring to FIG. 1A, an example of a system to transmit data is asystem 10. In one example, the system 10 is part of a windowedorthogonal frequency-division multiplexing (Wi-OFDM) system. The system10 includes a phase-shift keying (PSK) mapper 12, a frequency rotationmask processor 14, an inverse discrete Fourier transform (IDFT) module18, a time domain window module 22, a digital to analog converter (DAC)and a transmitter 28.

The PSK mapper 12 maps source bits, S_(B), to be transmitted. In oneexample, the output of the PSK mapper 12 is a complex data symbol,X_(k). After the PSK mapping, the frequency rotation mask processor 14applies a frequency rotation mask, e^(jθ) ^(k) , to the mapped data,where

${\theta_{k} = \frac{2{\pi ( {c + k} )}^{2}}{K}},$

where C is a constant and K is a randomness factor, k=0, 1, . . . ,N_(DFT)−1 and N_(DFT) is the size of the DFT and corresponds to thenumber of subcarriers. θ_(k) is an application of a polynomial phase(sometimes referred to as a polyphase) and is applied to eachsubcarrier. In one example, C is

$\frac{\pi}{2}.$

C can be adjusted by a user depending on the system 10 in order toachieve better performance. The randomness factor K controls symmetry ofthe frequency rotation mask defined by the polynomial phases. In oneexample, K is π². The output of the frequency rotation mask processor isX_(k)·e^(jθ) ^(k) .

The polynomial phase frequency rotation mask provides a set of phasesthat provide well behaved statistics, pulse compression or lowprobability of high PAPR/crest factor, and decoupling of energyconcentration in both the time domain and the frequency domain forarbitrary subcarrier locations and for all data sequences. Use of phasesthat decouple time domain and frequency domain provide a simpler methodcompared to iterative methods that go back and forth between time domainand frequency domain to reduce PAPR. Knowledge of well-behavedstatistics allows operations such as time domain windowing for spectrumcontainment and a tradeoff between clipping of an OFDM symbol in timedomain and ICI (Inter Carrier Interference).

The IDFT module 18 converts the frequency domain into the time domain.For example, the IDFT module 18 applies a term,

${\sum\limits_{k = 0}^{N_{DFT} - 1}\; ^{\frac{{j2\pi}\; {kn}}{N_{DFT}}}},$

where n=0, 1, . . . , N_(DFT)−1.

The output of the IDFT module 18 is:

$\sum\limits_{k = 0}^{N_{DFT} - 1}\; {X_{k} \cdot ^{{j\theta}_{k}} \cdot {^{\frac{{j2\pi}\; {kn}}{N_{DFT}}}.}}$

The time domain window module 22 applies a time window p(n). The outputof the time domain window 22 is an OFDM symbol sample, x(n), where:

${x(n)} = {{p(n)}{\sum\limits_{k = 0}^{N_{DFT} - 1}\; {X_{k} \cdot ^{{j\theta}_{k}} \cdot {^{\frac{{j2\pi}\; {kn}}{N_{DFT}}}.}}}}$

The OFDM symbol sample, x(n), is provided to the DAC 24 and converted toan analog signal, which is transmitted by the transmitter 28.

Referring to FIG. 1B, an example of a system for receiving datatransmitted from the system 10 is a system 30. In one example, thesystem 30 is part of a Wi-OFDM system. The system 30 includes a receiver32, an analog to digital converter (ADC) 34, a window inversion module38, a discrete Fourier transform (DFT) module 42, an inverse frequencyrotation mask processor 44 and a PSK de-mapper 48.

The receiver 32 receives the analog signal transmitted by the system 10and the analog signal is converted by the ADC 34 into digital data. Thewindow inversion module 38 removes the time domain window by applyingthe inverse of the term applied by the time-domain window module 22. Forexample, the window inversion module 38 applies the term p⁻¹(n).

The time domain data is converted by the DFT module 42 into thefrequency domain. The DFT module 42 applies an inverse of the termapplied by the IDFT module

18. For example, the DFT module 42 applies the term,

${\sum\limits_{n = 0}^{N_{DFT} - 1}\; ^{\frac{{j2\pi}\; {kn}}{N_{DFT}}}},$

to the output of the window inversion module 38.

The inverse frequency rotation mask processor 44 removes the frequencyrotation mask. The inverse frequency mask processor 44 applies aninverse of the term applied by the frequency rotation mask processor 14.For example, the inverse frequency mask processor 44 applies the term,e^(jθ) ^(k) .

The resultant data from the inverse frequency rotation mask processor 44is the complex symbol, X_(k), where:

$X_{k} = {( {\sum\limits_{n = 0}^{N_{DFT} - 1}\; {{p^{- 1}(n)} \cdot {x(n)} \cdot ^{\frac{{- {j2\pi}}\; {kn}}{N_{DFT}}}}} ) \cdot {^{- {j\theta}_{k}}.}}$

The complex symbol, X_(k), is provided to the PSK de-mapper 48 whichde-maps the data into the source bits, S_(B).

Referring to FIG. 2, an example of a process to transmit data is aprocess 200. Process 200 performs PSK mapping on source bits, S_(B)(202) and applies a frequency rotation mask (206). Process 200 performsan inverse discrete Fourier transform (210) and applies time window(212). Process 200 converts the digital signal into an analog signal(218) and transmits the analog signal (240).

Referring to FIG. 3, an example of a process to receive data is aprocess 300. Process 300 receives the analog signal (302) and convertsthe analog signal to a digital signal (306). Process 300 performs awindow inversion and performs a discrete Fourier transform (312).Process 300 applies an inverse frequency rotation mask (318) andperforms a PSK de-mapping (320) to recover the source bits, S_(B),processed by the system 10.

Referring to FIGS. 4A to 4C, by not applying a frequency rotation maskunacceptable signal dynamics or high PAPR signal characteristic appearsin the time domain due to coherent addition of subcarrier signals. Forexample, FIG. 4A is a graph of a signal in the frequency domain. In thetime domain, unwanted signals appear as large magnitude peaks such aspeaks 502 in the real part (FIG. 4B) and peaks 504 in the imaginary part(FIG. 4C). To correctly receive a signal, distortionless transmission isrequired. In order to avoid transmission-side distortion, the systemwill need to use expensive linear components on the transmitter-side. Inaddition, these peaks contain a majority of signal energy which timedomain windowing may significantly reduce.

Referring to FIGS. 4D to 4F, by applying a frequency rotation mask witha randomness factor, unwanted signal characteristics do not appear inthe time domain because each subcarrier is random enough such that thesubcarriers are added together incoherently. For example, FIG. 4D is agraph of a signal in the frequency domain, which is the same as FIG. 4A.In the time domain, the unwanted signal characteristics do not appear inthe real part (FIG. 4E) and in the imaginary part (FIG. 4F). As can beobserved from the associated figures, signal dynamics is lot moreconstrained and energy distribution is also greatly increased. Thisconstrained signal dynamics greatly reduce required linearity of atransmit-side system's components. At the same time, increased energydistribution minimizes transmit side energy reduction due to time domainwindowing.

Referring to FIGS. 5A to FIG. 5C, by a having randomness factor, thepolynomial phase frequency rotation mask is superior to other forms offrequency rotation masks. For example, FIG. 5A depicts using a Newmanfrequency rotation mask, where

${\theta_{k} = \frac{\pi \; k^{2}}{N_{DFT}}},$

and FIG. 5B depicts using a Narahashi-Nojima frequency rotation maskover the same

$\theta_{k} = {\frac{{\pi ( {k - 1} )}( {k - 2} )}{N_{DFT} - 1}.}$

subcarrier range as FIG. 5A, where The graphs in FIGS. 5A and 5B areapproximately the same and both figures illustrate the lack ofrandomness over a wide subcarrier range. Whereas the graph in FIG. 5C,which depicts a frequency rotation mask having a polynomial phase, ismore random over the same subcarrier range as FIGS. 5A and 5B becausethe θ_(k) in the polynomial phase includes the randomness factor K whichmake these phases asymmetrical or random for a given subcarrier range.

Referring to FIGS. 6A and 6B, the polynomial phases are superior to theother phases including the Newman phases. Note that these figures are infrequency domain and the Hamming window was used for a time-domainwindow operation. FIG. 6A is a graph that depicts sending a bit streamof all 1s using a Newman phases mask before a time-domain window andafter a time-domain window. After a time-domain window, there is areduction in amplitude of the signal. However, a graph in FIG. 6Bdepicts a bit stream of all 1s using a polynomial phase mask before atime-domain window and after a time-domain window, but showswell-behaved and acceptable loss of amplitude of the signal after thetime-domain window. All 1s sequence represents the worst case and it isobserved that the polynomial phases guarantee that the maximumsub-carrier to sub-carrier power deviation is within +/−5 dB for anyarbitrary data pattern. This is a significant improvement over +/−15 dBdeviation observed for the Newman phases.

Referring to FIG. 7, in one example, the system 10 and/or the system 30may be a computer such as a computer 700. The computer 700 includes aprocessor 702, a volatile memory 704, a non-volatile memory 706 (e.g.,hard disk) and the user interface (UI) 708 (e.g., a graphical userinterface, a mouse, a keyboard, a display, touch screen and so forth).The non-volatile memory 706 stores computer instructions 712, anoperating system 716 and data 718. In one example, the computerinstructions 712 are executed by the processor 702 out of volatilememory 704 to perform all or part of the processes described herein(e.g., processes 200 and 300).

The processes described herein (e.g., processes 200 and 300) are notlimited to use with the hardware and software of FIG. 7; they may findapplicability in any computing or processing environment and with anytype of machine or set of machines that is capable of running a computerprogram. The processes described herein may be implemented in hardware,software, or a combination of the two. The processes described hereinmay be implemented in computer programs executed on programmablecomputers/machines that each includes a processor, a non-transitorymachine-readable medium or other article of manufacture that is readableby the processor (including volatile and non-volatile memory and/orstorage elements), at least one input device, and one or more outputdevices. Program code may be applied to data entered using an inputdevice to perform any of the processes described herein and to generateoutput information.

The system may be implemented, at least in part, via a computer programproduct, (e.g., in a non-transitory machine-readable storage medium suchas, for example, a non-transitory computer-readable medium), forexecution by, or to control the operation of, data processing apparatus(e.g., a programmable processor, a computer, or multiple computers)).Each such program may be implemented in a high level procedural orobject-oriented programming language to communicate with a computersystem. However, the programs may be implemented in assembly or machinelanguage. The language may be a compiled or an interpreted language andit may be deployed in any form, including as a stand-alone program or asa module, component, subroutine, or other unit suitable for use in acomputing environment. A computer program may be deployed to be executedon one computer or on multiple computers at one site or distributedacross multiple sites and interconnected by a communication network. Acomputer program may be stored on a non-transitory machine-readablemedium that is readable by a general or special purpose programmablecomputer for configuring and operating the computer when thenon-transitory machine-readable medium is read by the computer toperform the processes described herein. For example, the processesdescribed herein may also be implemented as a non-transitorymachine-readable storage medium, configured with a computer program,where upon execution, instructions in the computer program cause thecomputer to operate in accordance with the processes. A non-transitorymachine-readable medium may include but is not limited to a hard drive,compact disc, flash memory, non-volatile memory, volatile memory,magnetic diskette and so forth but does not include a transitory signalper se.

The processes described herein are not limited to the specific examplesdescribed. For example, the processes 200 and 300 are not limited to thespecific processing order of FIGS. 2 and 3, respectively. Rather, any ofthe processing blocks of FIGS. 2 and 3 may be re-ordered, combined orremoved, performed in parallel or in serial, as necessary, to achievethe results set forth above.

The processing blocks (for example, in the processes 200 and 300)associated with implementing the system may be performed by one or moreprogrammable processors executing one or more computer programs toperform the functions of the system. All or part of the system may beimplemented as, special purpose logic circuitry (e.g., an FPGA(field-programmable gate array) and/or an ASIC (application-specificintegrated circuit)). All or part of the system may be implemented usingelectronic hardware circuitry that include electronic devices such as,for example, at least one of a processor, a memory, a programmable logicdevice or a logic gate.

Elements of different embodiments described herein may be combined toform other embodiments not specifically set forth above. Otherembodiments not specifically described herein are also within the scopeof the following claims.

What is claimed is:
 1. A method comprising: performing a mapping on bitsto form a complex data symbol; applying a frequency rotation mask to thecomplex data symbol based on a polynomial phase; performing an inversediscrete Fourier transform (IDFT) after applying the frequency rotationmask; applying a time domain window after performing the IDFT;converting digital data to analog data after applying the time window;and transmitting the analog data as an analog signal.
 2. The method ofclaim 1, further comprising: receiving the analog signal; converting theanalog signal to a digital signal; performing a window inversion on thedigital signal; performing a discrete Fourier transform (DFT) to formtime domain data after performing the window inversion; applying aninverse of the frequency rotation mask on the time domain data to formthe complex data symbol; and performing a demapping to convert thecomplex data symbol into the bits.
 3. The method of claim 1 whereinapplying a frequency rotation mask to the complex data symbol based onpolynomial phases comprises applying a frequency rotation mask to thecomplex data symbol based on a polynomial phase comprising a randomnessfactor.
 4. The method of claim 3 wherein applying, to the mapped sourcebits, a frequency rotation mask based on polynomial phases comprising arandomness factor comprises applying a frequency rotation mask to thecomplex data symbol based on a polynomial phase comprising a randomnessfactor of π².
 5. The method of claim 1 wherein performing a mapping onbits to form a complex data symbol comprises performing phase-shiftkeying (PSK) mapping on bits to form a complex data symbol.
 6. Themethod of claim 1 wherein applying, to the mapped source bits, afrequency rotation mask based on polynomial phases comprises applying afrequency rotation mask to the complex data symbol based on a polynomialphase comprising e^(jθ) ^(k) , where${\theta_{k} = \frac{2{\pi ( {c + k} )}^{2}}{K}},$ C is aconstant and K is a randomness factor, k=0, 1, . . . , N_(DFT)−1 andN_(DFT) is the size of a discrete Fourier transform.
 7. An apparatus,comprising: electronic hardware circuitry configured to: perform amapping on bits to form a complex data symbol; apply a frequencyrotation mask to the complex data symbol based on a polynomial phase;perform an inverse discrete Fourier transform (IDFT) after applying thefrequency rotation mask; apply a time domain window after performing theIDFT; convert digital data to analog data after applying the timewindow; and transmit the analog data as an analog signal.
 8. Theapparatus of claim 7 wherein the circuitry comprises at least one of aprocessor, a memory, a programmable logic device or a logic gate.
 9. Theapparatus of claim 7, further comprising circuitry configured to:receive the analog signal; convert the analog signal to a digitalsignal; perform a window inversion on the digital signal; perform adiscrete Fourier transform (DFT) to form time domain data afterperforming the window inversion; apply an inverse of the frequencyrotation mask on the time domain data to form the complex data symbol;and perform a demapping to convert the complex data symbol into thebits.
 10. The apparatus of claim 7 wherein the circuitry to apply afrequency rotation mask to the complex data symbol based on polynomialphases comprises circuitry configured to apply a frequency rotation maskto the complex data symbol based on a polynomial phase comprising arandomness factor.
 11. The apparatus of claim 10 wherein the circuitryto apply, to the mapped source bits, a frequency rotation mask based onpolynomial phases comprising a randomness factor comprises circuitryconfigured to apply a frequency rotation mask to the complex data symbolbased on a polynomial phase comprising a randomness factor of π². 12.The apparatus of claim 7 wherein the circuitry to perform a mapping onbits to form a complex data symbol comprises circuitry configured toperform phase-shift keying (PSK) mapping on bits to form a complex datasymbol.
 13. The apparatus of claim 7 wherein the circuitry to apply, tothe mapped source bits, a frequency rotation mask based on polynomialphases comprises circuitry configured to apply a frequency rotation maskto the complex data symbol based on a polynomial phase comprising e^(jθ)^(k) , where${\theta_{k} = \frac{2{\pi ( {c + k} )}^{2}}{K}},$ C is aconstant and K is a randomness factor, k=0, 1, . . . , N_(DFT)−1 andN_(DFT) is the size of a discrete Fourier transform.
 14. An articlecomprising: a non-transitory computer-readable medium that storescomputer-executable instructions, the instructions causing a machine to:perform a mapping on bits to form a complex data symbol; apply afrequency rotation mask to the complex data symbol based on a polynomialphase; perform an inverse discrete Fourier transform (IDFT) afterapplying the frequency rotation mask; apply a time domain window afterperforming the IDFT; convert digital data to analog data after applyingthe time window; and transmit the analog data as an analog signal. 15.The article of claim 14, further comprising computer-executableinstructions causing the machine to: receive the analog signal; convertthe analog signal to a digital signal; perform a window inversion on thedigital signal; perform a discrete Fourier transform (DFT) to form timedomain data after performing the window inversion; apply an inverse ofthe frequency rotation mask on the time domain data to form the complexdata symbol; and perform a demapping to convert the complex data symbolinto the bits.
 16. The article of claim 14 wherein the instructionscausing the machine to apply a frequency rotation mask to the complexdata symbol based on polynomial phases comprises instructions causingthe machine to apply a frequency rotation mask to the complex datasymbol based on a polynomial phase comprising a randomness factor. 17.The article of claim 16 wherein the instructions causing the machine toapply, to the mapped source bits, a frequency rotation mask based onpolynomial phases comprising a randomness factor comprises instructionscausing the machine to apply a frequency rotation mask to the complexdata symbol based on a polynomial phase comprising a randomness factorof π².
 18. The article of claim 14 wherein the instructions causing themachine to perform a mapping on bits to form a complex data symbolcomprises instructions causing the machine to perform phase-shift keying(PSK) mapping on bits to form a complex data symbol.
 19. The article ofclaim 14 wherein the instructions causing the machine to apply, to themapped source bits, a frequency rotation mask based on polynomial phasescomprises instructions causing the machine to apply a frequency rotationmask to the complex data symbol based on a polynomial phase comprisinge^(jθ) ^(k) , where${\theta_{k} = \frac{2{\pi ( {c + k} )}^{2}}{K}},$ C is aconstant and K is a randomness factor, k=0, 1, . . . , N_(DFT)−1 andN_(DFT) is the size of a discrete Fourier transform.